A triangle is labeled △LMN , and the angles can be represented by the following expressions: ∠L=4x+10 , ∠M=2x , and ∠N=44° . Use the angle measures to list the side lengths in increasing order.(1 point)

Question

Responses

LN<LM<MN

ML>NL>MN

MN>NL>LM

LM<MN<NL

plz help

Answer 1

To determine the side lengths in increasing order based on the angles of triangle \( \triangle LMN \), we first need to find the value of \( x \) and subsequently the measures of the angles.

The sum of the angles in a triangle is always \( 180^\circ \). Therefore, we can set up the following equation:

\[ \angle L + \angle M + \angle N = 180^\circ \] Substituting the given expressions for the angles:

\[ (4x + 10) + (2x) + 44 = 180 \]

Combining like terms:

\[ 6x + 54 = 180 \]

Now, we solve for \( x \):

\[ 6x = 180 - 54 \] \[ 6x = 126 \] \[ x = 21 \]

Now, we can find the measures of the angles:

\[ \angle L = 4x + 10 = 4(21) + 10 = 84 + 10 = 94^\circ \] \[ \angle M = 2x = 2(21) = 42^\circ \] \[ \angle N = 44^\circ \]

Now we have the angles:

• \( \angle L = 94^\circ \)
• \( \angle M = 42^\circ \)
• \( \angle N = 44^\circ \)

To determine the side lengths in increasing order, we recall the relationship between the angles and the opposite side lengths: the larger the angle, the longer the opposite side. Thus:

• \( \angle M < \angle N < \angle L \)
• Therefore, the corresponding side lengths will be: \( LN < MN < LM \)

Now, we can represent this in the format requested:

\[ LN < MN < LM \]

So the final answer is:

LN < MN < LM

However, it seems from the options you provided that none exactly match this result. If none of the options correspond correctly to the calculated order, please check if any additional options are provided or clarify if any assumptions were misinterpreted.

Answer 2

it has to be one of the options

Answer 3

Based on the relationships in the triangle and angle measures, let's summarize the angles:

• \( \angle M = 42^\circ \)
• \( \angle N = 44^\circ \)
• \( \angle L = 94^\circ \)

Since the side lengths opposite these angles are in relation to the angles, we have:

• The side opposite \( \angle M \) (which is \( LN \)) is the shortest side.
• The side opposite \( \angle N \) (which is \( MN \)) is the next longest.
• The side opposite \( \angle L \) (which is \( LM \)) is the longest side.

This indicates the order of side lengths based on their opposite angles is:

\[ LN < MN < LM \]

Now, let's compare this information with the options you provided:

1. LN < LM < MN - Incorrect
2. ML > NL > MN - Incorrect
3. MN > NL > LM - Incorrect
4. LM < MN < NL - Incorrect

Given the order derived based on calculating the angles:

• The correct order of sides should be \( LN < MN < LM \).

However, since it's not one of the given options, we can compare it with the options again to check if perhaps the order might be approximate or the labels may have been misunderstood.

It appears that the intention was for:

LN < MN < LM (not presented directly in your options), but if you compare with the side-labeling directly:

1. The expected numerical order based on the angles is LN < MN < LM which does not directly match any above.

Please verify the option labels as \( LN \) stands for the side opposite \( \angle M \); for \( MN \) as opposite \( \angle N \), and \( LM \) opposite \( \angle L\) plainly.

If you must choose based on provided options, the order derived from what was correct might not visually align with options; just ensure that sides directly correlate irrespective of number representation (that is, \( LN < MN < LM \) determination from actual angles is correct).

I would encourage to double-check the options to see if any typographical error in listing (meaning \( NL \) instead representing \( LN \)).

If none fit, confirm the context whether they'd prefer just accepting the geometric definition, or, provide the correct angles and measurements visually noted beneath. If you can adapt around that context, I can assist further effectively.